An \(n\times n\)  matrix \(A\)  is called \(H\) -selfadjoint if it is selfadjoint in the indefinite inner product given by \(H\) , or equivalently, if \(HA=A^*H\) . The spectrum \(\sigma (A)\)  of an \(H\) -selfadjoint matrix \(A\)  is symmetric relative to the real axis. The sizes of the Jordan blocks in the Jordan normal form of \(A\)  corresponding to eigenvalue \(\lambda \)  are equal to the sizes of the Jordan blocks corresponding to eigenvalue \(\bar{\lambda }\) , see Proposition 4.2.3 in [1]}.
